2.2 Chandrasekhar–Sasaki–Nakamura transformation

As seen from the asymptotic behaviors of the radial functions given in Equations (24View Equation) and (25View Equation), the Teukolsky equation is not in the form of a canonical wave equation near the horizon and infinity. Therefore, it is desirable to find a transformation that brings the radial Teukolsky equation into the form of a standard wave equation.

In the Schwarzschild case, Chandrasekhar found that the Teukolsky equation can be transformed to the Regge–Wheeler equation, which has the standard form of a wave equation with solutions having regular asymptotic behaviors at horizon and infinity [21Jump To The Next Citation Point]. The Regge–Wheeler equation was originally derived as an equation governing the odd parity metric perturbation [87Jump To The Next Citation Point]. The existence of this transformation implies that the Regge–Wheeler equation can describe the even parity metric perturbation simultaneously, though the explicit relation of the Regge–Wheeler function obtained by the Chandrasekhar transformation with the actual metric perturbation variables has not been given in the literature yet.

Later, Sasaki and Nakamura succeeded in generalizing the Chandrasekhar transformation to the Kerr case [92Jump To The Next Citation Point, 93Jump To The Next Citation Point]. The Chandrasekhar–Sasaki–Nakamura transformation was originally introduced to make the potential in the radial equation short-ranged, and to make the source term well-behaved at the horizon and at infinity. Since we are interested only in bound orbits, it is not necessary to perform this transformation. Nevertheless, because its flat-space limit reduces to the standard radial wave equation in the Minkowski spacetime, it is convenient to apply the transformation when dealing with the post-Minkowski or post-Newtonian expansion, at least at low orders of expansion.

We transform the homogeneous Teukolsky equation to the Sasaki–Nakamura equation [92, 93], which is given by

( 2 ) -d--− F(r) -d-− U(r) X ℓmω = 0. (50 ) dr∗2 dr∗
The function F(r) is given by
η Δ F(r) = --,r-------, (51 ) η r2 + a2
where
c1 c2 c3 c4 η = c0 + -- + -2 + --3 + -4, (52 ) r r r r
with
c0 = − 12iωM + λ(λ + 2) − 12a ω(aω − m ), c1 = 8ia[3a ω − λ(aω − m )], c = − 24iaM (aω − m ) + 12a2[1 − 2(aω − m )2], 2 (53 ) c3 = 24ia3(aω − m ) − 24M a2, c4 = 12a4.
The function U(r) is given by
--ΔU1----- 2 -ΔG,r-- U (r) = (r2 + a2)2 + G + r2 + a2 − FG, (54 )
where
2-(r-−-M--) ---rΔ----- G = − r2 + a2 + (r2 + a2)2, (55 ) [( ) ( ) ] Δ2- β,r η,r β,r U1 = V + β 2α + Δ − η α + Δ , (56 ) ,r K-β- 6Δ- α = − i Δ2 + 3iK,r + λ + r2 , (57 ) ( 2Δ ) β = 2 Δ − iK + r − M − --- . (58 ) r

The relation between R ℓm ω and X ℓmω is

[( ) ] 1 β,r β R ℓm ω = η- α + Δ-- χℓmω − Δ-χℓmω,r , (59 )
where χℓmω = X ℓmω Δ∕ (r2 + a2)1∕2. Conversely, we can express X ℓmω in terms of R ℓm ω as
( ) X ℓmω = (r2 + a2)1∕2 r2J− J− 1-R ℓm ω , (60 ) r2
where J− = (d∕dr ) − i(K ∕Δ ).

If we set a = 0, this transformation reduces to the Chandrasekhar transformation for the Schwarzschild black hole [21]. The explicit form of the transformation is

Δ ( d ) r2 ( d ) R ℓm ω = -- ----+ iω -- ----+ iω rX ℓmω, (61 ) c0 ( dr∗ ) Δ ( dr∗ ) r5 d r2 d Rℓmω X ℓm ω = -- --∗-− iω -- --∗-+ iω --2--, (62 ) Δ dr Δ dr r
where c0, defined in Equation (53View Equation), reduces to c0 = (ℓ − 1)ℓ(ℓ + 1 )(ℓ + 2) − 12iM ω. In this case, the Sasaki–Nakamura equation (50View Equation) reduces to the Regge–Wheeler equation [87], which is given by
( d2 ) --∗2 + ω2 − V (r) Xℓω(r) = 0, (63 ) dr
where
( 2M ) ( ℓ(ℓ + 1) 6M ) V (r) = 1 − ---- ----2---− --3- . (64 ) r r r
As is clear from the above form of the equation, the lowest order solutions are given by the spherical Bessel functions. Hence it is intuitively straightforward to apply the post-Newtonian expansion to it. Some useful techniques for the post-Newtonian expansion were developed for the Schwarzschild case by Poisson [83Jump To The Next Citation Point] and Sasaki [91Jump To The Next Citation Point].

The asymptotic behavior of the ingoing wave solution Xin which corresponds to Equation (19View Equation) is

{ Aref eiωr∗ + Ainc e− iωr∗ for r∗ → ∞, Xinℓmω → ℓmω ∗ ℓmω (65 ) Atrℓamnωse−ikr for r∗ → − ∞.
The coefficients Ainc, Aref, and Atrans are related to Binc, Bref, and Btrans, defined in Equation (19View Equation), by
1 Binℓmcω = − ---2Aiℓnmcω, (66 ) 4ω 2 Bref = − 4ω--Aref , (67 ) ℓm ω c0 ℓm ω 1 Btℓrmanωs = -----Atrℓmanωs , (68 ) dℓm ω
where
------ d = ∘ 2M r [(8 − 24iM ω − 16M 2ω2)r2 ℓm ω + + 2 + (12iam − 16M + 16amM ω + 24iM ω )r+ − 4a2m2 − 12iamM + 8M 2]. (69 )
In the following sections, we present a method of post-Newtonian expansion based on the above formalism in the case of the Schwarzschild background. In the Kerr case, although a post-Newtonian expansion method developed in previous work [94Jump To The Next Citation Point, 101Jump To The Next Citation Point] was based on the Sasaki–Nakamura equation, we will not present it in this paper. Instead, we present a different formalism, namely the one developed by Mano, Suzuki, and Takasugi which allows us to solve the Teukolsky equation in a more systematic manner, albeit very mathematical [68Jump To The Next Citation Point]. The reason is that the equations in the Kerr case are already complicated enough even if one uses the Sasaki–Nakamura equation, so that there is not much advantage in using it. In contrast, in the Schwarzschild case, it is much easier to obtain physical insight into the role of relativistic corrections if we deal with the Regge–Wheeler equation.


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